分支隨機遊走和對數相關場的極值

Abstract

這篇論文是探討對數相關場的最大值的行為。隨機場中最大值的行為在物理學、機率論和數論等各個領域都具有重要意義。在本論文中，我們將重點放在分支隨機遊走（BRW）和二維離散高斯自由場（GFF）。我們將證明 BRW 最大值的緊密性和收斂性，並建立 GFF 和 BRW 之間的連結。然後我們會給兩個對數相關場的例子：臨界線上的黎曼 zeta 函數和隨機酉矩陣的特徵多項式。

This thesis explores the behavior of the maximum of log-correlated fields. The be- havior of maxima in random fields holds significant importance in various fields, includ- ing physics, probability theory, and number theory. In this thesis, we focus on branching random walks (BRW) and two dimensional discrete Gaussian free fields (GFF). We will prove the tightness and convergence in law of the centered maximum for BRW and estab- lish a connection between GFF and BRW. Then we present two examples of log-correlated fields, namely the Riemann zeta function on the critical line and the characteristic poly- nomial of a random unitary matrix.